It has been more than a year since a five-judge appeals court first heard arguments on whether a woman who conceived a baby with a stranger's sperm in a fertility treatment mix-up is entitled to damages for bringing up the child. But no answer is in sight to the knotty question.

A second hearing yesterday continued to explore various legal issues raised in the case, described as "unique" and "difficult" by the judges, and ended without a conclusion.

The three-hour discussion touched not only on the woman's claim for upkeep of the child, but also widened to other possible categories of damages she had not pleaded, including punitive damages and damages for loss of autonomy.

The conundrum arose in a negligence suit the woman filed in 2012 against Thomson Fertility Centre, its parent company Thomson Medical, and two embryologists.

In 2010, the woman and her husband went to the centre for in-vitro fertilisation treatment, but a stranger's sperm - instead of her husband's - was used to fertilise her extracted eggs.

The mistake resulted in her giving birth to a baby girl, now six years old, with her genetic make-up but not her husband's.

The couple's suspicions were first aroused by the newborn's complexion and later by her blood type.

In her suit, the woman sought damages for various categories of claims, including for the upkeep of the child, known as Baby P.

They included expenses for necessities until the child is financially self-reliant, education in Beijing where the family was based, and tertiary education in Germany, the husband's home country.

Ahead of the assessment of damages, the defendants asked the High Court to rule on whether Singapore law allows damages to be awarded for the expense of raising a healthy child.

Last year, the High Court disallowed the claim, citing policy considerations which view the birth of a healthy child as a blessing. The woman appealed. The appeal was first heard last August.

In written arguments filed in May, Associate Professor Goh Yihan from the Singapore Management University, who was appointed to give an independent view, said any negative connotation associated with upkeep costs can be avoided by recharacterising the claim as a loss of autonomy, shifting the focus from the child to the parents' lives.

As the hearing continued yesterday, Chief Justice Sundaresh Menon questioned if the mental harm the woman suffered could be compensated and, if so, how it should be quantified. He wondered aloud if upkeep costs could be seen as a proxy for damages for emotional distress, offset against the joys of having a child.

The woman's lawyer, Senior Counsel N. Sreenivasan, argued against such a weighing exercise, saying the boon and bane of raising the child are "incommensurable" and "two different sides of the same problem". He said his client still has emotional anguish in raising the child.

The Chief Justice also wondered if the court should impose punitive damages, awarded to punish defendants for their conduct. Senior Counsel Lok Vi Ming, for the defendants, argued that that amounted to double punishment, as Thomson Medical was fined the maximum $20,000 for regulatory offences in 2011.